Question

Given that \( \cot \left( \frac{A+B}{2} \right) \cdot \tan \left( \frac{A-B}{2} \right) = \), and the equation \( \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \), find the area of \( \Delta ABC = 2 \).

• A. \( 2 \)
• B. \( 3 \)
• C. \( 4 \)
• D. \( 5 \)

Hint:

When given multiple types of equations (trigonometric and coordinate), try simplifying each to extract relevant information. In this case, the area was directly provided.

Answer 1

The Correct Option is A

This is a geometric problem combining trigonometry and coordinate geometry.
Step 1: Analyze the given equations The provided equation is: \[ \frac{x}{2} + \frac{y}{3} + \frac{2}{6} - 1 = 0 \] Simplified equation: \[ \frac{x}{2} + \frac{y}{3} + \frac{1}{3} = 1 \] Multiplying by 6 to eliminate denominators: \[ 3x + 2y + 2 = 6 \] Further simplification yields: \[ 3x + 2y = 4 \] This represents a linear equation in \( x \) and \( y \), corresponding to a line in the coordinate plane.
Step 2: Relating trigonometric functions to area The trigonometric equation is given as: \[ \cot \left( \frac{A+B}{2} \right) \cdot \tan \left( \frac{A-B}{2} \right) \] This expression likely relates to the angles of a triangle. Such relationships are often used to determine a triangle's area via trigonometric identities. However, the area of \( \Delta ABC \) is already stated as \( 2 \), obviating the need for explicit calculation.
Step 3: Conclusion Therefore, the area of triangle \( \Delta ABC \) is \( \boxed{2} \).